Simplify (1 - sin^2 x)/cos x

Step by step

The question

Question: simplify (1 − sin² x) over cos x.

Simplify means: write it as simply as we can.

1 Use sin² + cos² = 1

The most important identity: sin² x + cos² x = 1.

It comes from the unit circle.

Moved around, it says 1 − sin² x = cos² x.

Careful!

The square stays: 1 − sin² x is cos² x, with the square.

So we put cos² x in place of (1 − sin² x).

2 Cancel

A square means times itself: cos² x = cos x × cos x.

We write every one out.

Every cos x on top has a partner at the bottom.

Each pair over itself is 1.

Cross them out.

What is left: cos x.

So it simplifies to cos x.

3 Check

Let's check with a number: x = 30°.

On the calculator, (1 − sin² 30°) over cos 30° is about 0.8660.

And cos 30° is about 0.8660.

The same!

So it is right.

Let's look back.

First: use sin² + cos² = 1.

Then: cancel.

Last: check.